Four-genus and four-dimensional clasp number of a knot

Murakami, Hitoshi; Yasuhara, Akira

doi:10.1090/S0002-9939-00-05461-7

Four-genus and four-dimensional clasp number of a knot
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by Hitoshi Murakami and Akira Yasuhara PDF

Proc. Amer. Math. Soc. 128 (2000), 3693-3699 Request permission

Abstract:

For a knot $K$ in the $3$-sphere, by using the linking form on the first homology group of the double branched cover of the $3$-sphere, we investigate some numerical invariants, $4$-genus $g^*(K)$, nonorientable $4$-genus $\gamma ^*(K)$ and $4$-dimensional clasp number $c^*(K)$, defined from the four-dimensional viewpoint. T. Shibuya gave an inequality $g^*(K)\leq c^*(K)$, and asked whether the equality holds or not. From our result in this paper, we find that the equality does not hold in general.

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